Elliott was born and raised in Baltimore, Maryland. He originally planned to become a studio musician and studied organ as part of this goal. After studies at the preparatory division of the Peabody Conservatory and the Catholic University of America, he received a bachelor's degree from the Curtis Institute of Music while also serving as an assistant organist for the Wanamaker Organ. It was while he was a student at Curtis that Elliott joined the LDS Church. He served a mission for the LDS Church in Argentina from 1981 to 1983.
Elliott received Master of Music and Doctor of Musical Arts degrees from the Eastman School of Music, studying under David Craighead. He then became a professor of music at Brigham Young University. He was appointed a Tabernacle organist when Robert Cundick retired in 1991. In addition to accompanying the Tabernacle Choir and giving recitals at Temple Square, Elliott has recorded many organ pieces with various labels and occasionally gives organ recitals at various locations across the United States. Several of Elliott's arrangements for organ have been published, many by Jackman Music.